
\begin{section}{Complex Functions}

\begin{subsection}{Limits and Continuity of Functions on the Complex Plane}

A {\bf complex-valued function} $f$ is a function which maps each complex number $z$ in its domain $D$ to one and only one complex number $w$. We write $w = f(z)$ and call $w$ the {\bf image} of $z$ under $f$. The set of all such images $\{ w = f(z) \colon z \in D \}$ is called the {\bf range} of $f$.

Metaphorically speaking -- the domain of a function is the set of all arrows in a quiver, the image is the target to shoot at, and the range is all of the points on the target which are hit by an arrow. 

A complex-valued function can be decomposed into its real and imaginary components, similar to what we can do with a complex number.

$$w = f(z) = f(x,y) = f(x + iy) = u + iv.$$

Since the values of $u$ and $v$ depend on $x$ and $y$ (the real and imaginary parts of $z$), we can think of them as real-valued functions of two variables. That is,

$$u = u(x,y), v = v(x,y).$$

Combining the two equations above gives a convenient way to work with complex-valued functions, which we can now write as:

\begin{equation}
f(z) = f(x + iy) = u(x,y) + iv(x,y).
\end{equation}

\begin{ex}
Consider the function $f(z) = z^{4}$, which can be rewritten in the form $f(z) = u(x,y) + iv(x,y)$. That is,
$$f(z) = (x + iy)^{4} = x^{4} + 4x^{3}iy + 6x^{2}(iy)^{2} + 4x(iy)^{3} + (iy)^{4} = (x^{4} - 6x^{2}y^{2}+ y^{4}) + i(4x^{3}y - 4xy^{3}) = u(x,y) + iv(x,y).$$

\end{ex}

We can also decompose a function in polar form into component real-valued functions. Let $z = re^{i\theta}$, then

\begin{equation}
f(z) = f(re^{i\theta}) = u(r,\theta) + iv(r, \theta).
\end{equation}

\begin{ex}
Consider the function $f(z) = z^{2}$, which can be rewritten in the form $f(z) = u(r,\theta) + iv(r, \theta)$. That is,
$$f(re^{i\theta}) = (re^{i\theta})^{2} = r^{2}e^{2i\theta} = r^{2}\cos(2\theta) + i\sin(2\theta) = u(r,\theta) + iv(r,\theta).$$
\end{ex}

\end{subsection}

\begin{subsection}{The Mappings $w = z^{n}$ and $w = z^{\frac{1}{n}}$}

\end{subsection}

\begin{subsection}{Branch Cuts of Functions}

\end{subsection}

\begin{subsection}{The Reciprocal Transformation $w=\frac{1}{z}$}

\end{subsection}

\end{section}

